AbstractSolutions to nonlinear Schrödinger equations may blow up in finite
time. We study the influence of the introduction of a potential on this
phenomenon. For a linear potential (Stark effect), the blow-up time
remains unchanged, but the location of the collapse is altered. The main
part of our study concerns isotropic quadratic potentials. We show that
the usual (confining) harmonic potential may anticipate the blow-up
time, and always does when the power of the nonlinearity is
L^2-critical. On the other hand, introducing a "repulsive" harmonic
potential prevents finite time blow-up, provided that this potential is
sufficiently "strong". For the L^2-critical nonlinearity, this
mechanism is explicit: according to the strength of the potential,
blow-up is first delayed, then prevented.